Actually I do not have a great appreciation of this. The wikipedia page seems to suggest that a forward-backward asymmetry is intended to probe the differences between left and right handed fermions, however since the only way to have left handed particles in one beam and right handed in another beam is to use particle/antiparticle pairs (i.e. electron/positron like LEP, or proton/antiproton like the Tevatron) then I have a hard time understanding what information it gives you that charge asymmetry measurements do not. The wikipedia page does mention some things but I still don't really get it. It is pretty technical I think.
Perhaps someone will come along and educate us both :)